Scilab Function
Last update : 12/1/2004

systems - a collection of dynamical system

### Calling Sequence

  `[]=systems()`

### Description

A call to this function will load into Scilab a set of macros which describes dynamical systems. Their parameters can be initiated by calling the routine tdinit().

### Bioreact

    [ydot]=biorecat(t,x)

a bioreactor model,

  - x(1)is the biomass concentration
  - x(2)is the sugar concentration

                        xdot(1)=mu_td(x(2))*x(1)- debit*x(1);
                        xdot(2)=-k*mu_td(x(2))*x(1)-debit*x(2)+debit*x2in;

where mu_td is given by

                        mu_td(x)=x/(1+x);

### Compet

    [xdot]=compet(t,x [,u])

a competition model. `x(1),x(2)` stands for two populations which grows on a same resource. `1/u` is the level of that resource ( 1 is the default value).

    xdot=0*ones(2,1);
    xdot(1) = ppr*x(1)*(1-x(1)/ppk) - u*ppa*x(1)*x(2) ,
    xdot(2) = pps*x(2)*(1-x(2)/ppl) - u*ppb*x(1)*x(2) ,

"The macro `[xe]=equilcom(ue)`" computes an equilibrium point of the competition model and a fixed level of the resource ue ( default value is 1)

"The macro `[f,g,h,linsy]=lincomp([ue])`" gives the linearisation of the competition model ( with output y=x) around the equilibrium point xe=equilcom(ue). This macro returns [f,g,h] the three matrices of the linearised system. and linsy which is a Scilab macro [ydot]=linsy(t,x) which computes the dynamics of the linearised system

### Cycllim

    [xdot]=cycllim(t,x)

a model with a limit cycle

      xdot=a*x+qeps(1-||x||**2)x

### Linear

    [xdot]=linear(t,x)

a linear system

### Blinper

    [xdot]=linper(t,x)

a linear system with quadratic perturbations.

### Pop

    [xdot]=pop(t,x)

a fish population model

    xdot= 10*x*(1-x/K)- peche(t)*x

### Proie

a Predator prey model with external insecticide.

    [xdot]=p_p(t,x,[u]

  - x(1)The prey ( that we want to kill )
  - x(2)the predator ( that we want to preserve )
  - umortality rate due to insecticide which destroys both prey and predator ( default value u=0)

    xdot(1) = ppr*x(1)*(1-x(1)/ppk) - ppa*x(1)*x(2) - u*x(1);
    xdot(2) = -ppm*x(2)             + ppb*x(1)*x(2) - u*x(2);

The macro `[xe]=equilpp([ue])` computes the equilibrium point of the `p_p` system for the value `ue` of the command. The default value for `ue` is 0.

                        xe(1) =  (ppm+ue)/ppb;
                        xe(2) =  (ppr*(1-xe(1)/ppk)-ue)/ppa;

### Lincom

    [xdot]=lincom(t,x,k)

linear system with a feedback

     xdot= a*x +b*(-k*x)

### See Also

`tdinit`,